Theorem 0nnq 10943

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5693	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10931	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4062	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3959	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109  ∀wral 3052  ∅c0 4313   class class class wbr 5124   × cxp 5657  ‘cfv 6536  2^nd c2nd 7992  Ncnpi 10863   <_N clti 10866   ~_Q ceq 10870  Qcnq 10871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-xp 5665  df-nq 10931

This theorem is referenced by:  adderpq  10975  mulerpq  10976  addassnq  10977  mulassnq  10978  distrnq  10980  recmulnq  10983  recclnq  10985  ltanq  10990  ltmnq  10991  ltexnq  10994  nsmallnq  10996  ltbtwnnq  10997  ltrnq  10998  prlem934  11052  ltaddpr  11053  ltexprlem2  11056  ltexprlem3  11057  ltexprlem4  11058  ltexprlem6  11060  ltexprlem7  11061  prlem936  11066  reclem2pr  11067